As I have always thought that a radius is the smallest "component" in an image (i.e - it can only be one colour or tone), I am confused by the reference to "fractions of a pixel" and how radius can affect a partial pixel. Can someone explain?

By default, the radius in ACR/LR is 1 pixel. That means the sharpening algorithm will apply the sharpening to edges based on the 1 pixel radius. ACR/LR can determine the sharpening based on a radius from .5 pixels to 3 pixels. When calculating the radius below 1 pixel it does indeed use a fraction. It makes the sharpening super fine.